Physical Review A

Shear viscosity of the normal phase of a unitary Fermi gas from the ɛ expansion

Andrei B. Kryjevski

Published Feb 14, 2014• 3 citations

Abstract

Using the $\ensuremath{\varepsilon}$-expansion technique, we compute $\ensuremath{\eta}/s$, where $\ensuremath{\eta}$ is the shear viscosity and $s$ is the entropy density of the normal phase of unitary Fermi gas in $d=4\ensuremath{-}\ensuremath{\varepsilon}$ dimensions to leading order (LO) in $\ensuremath{\varepsilon}$. We use the kinetic theory approach and solve transport equations for a medium perturbed by a shear hydrodynamic flow. The collision integrals are calculated to ${\ensuremath{\varepsilon}}^{2}$, which is LO. The LO result is temperature independent with $\ensuremath{\eta}/s\ensuremath{\simeq}(0.11/{\ensuremath{\varepsilon}}^{2})(\ensuremath{\hbar}/{k}_{B})$. The $d=3$ prediction for $\ensuremath{\eta}/s$ exceeds the $\ensuremath{\hbar}/4\ensuremath{\pi}{k}_{B}$ bound by a factor of about $1.4$.

DOI: 10.1103/physreva.89.023621

Related papers

$η/s$ of the Normal Phase of Unitary Fermi Gas from $\varepsilon$ Expansion

Andrei B. Kryjevski

arXiv:1206.0059Jun 1, 20120 citations

Determination of the Density and Temperature Dependence of the Shear Viscosity of a Unitary Fermi Gas Based on Hydrodynamic Flow

Marcus Bluhm, Jiaxun Hou, Thomas Schäfer

arXiv:1704.03720Physical Review LettersAug 10, 201736 citations

Nearly Perfect Fluidity in the Dilute Fermi Gas: An Update

Thomas Schäfer

In this contribution we summarize recent results on the transport properties of strongly correlated dilute Fermi gases. We discuss the hydrodynamic equations in the normal phase and present new results on the structure of second order terms in the gradient expansion. We also discuss predictions from kinetic theory, and show how these predictions can be tested using experimental data on elliptic flow. We summarize current constraints on the shear viscosity to entropy density ratio η/s.

Aug 16, 20160 citations

Viscosity and scale invariance in the unitary Fermi gas

Tilman Enss, R. Haussmann, W. Zwerger

Jan 20, 2016164 citations

Hydrodynamic fluctuations and the minimum shear viscosity of the dilute Fermi gas at unitarity

Clifford Chafin, Thomas Schäfer

arXiv:1209.1006Physical Review AFeb 27, 201351 citations

Medium effects and the shear viscosity of the dilute Fermi gas away from the conformal limit

Marcus Bluhm, Thomas Schäfer

arXiv:1410.2827Physical Review ADec 8, 201411 citations

Physical Review A

Shear viscosity of the normal phase of a unitary Fermi gas from the ɛ expansion

Andrei B. Kryjevski

Published Feb 14, 2014• 3 citations

Publisher

Abstract

Using the

\ensuremath{\varepsilon}

-expansion technique, we compute

\ensuremath{\eta}/s

, where

\ensuremath{\eta}

is the shear viscosity and

s

is the entropy density of the normal phase of unitary Fermi gas in

d=4\ensuremath{-}\ensuremath{\varepsilon}

dimensions to leading order (LO) in

\ensuremath{\varepsilon}

. We use the kinetic theory approach and solve transport equations for a medium perturbed by a shear hydrodynamic flow. The collision integrals are calculated to

{\ensuremath{\varepsilon}}^{2}

, which is LO. The LO result is temperature independent with

\ensuremath{\eta}/s\ensuremath{\simeq}(0.11/{\ensuremath{\varepsilon}}^{2})(\ensuremath{\hbar}/{k}_{B})

. The

d=3

prediction for

\ensuremath{\eta}/s

exceeds the

\ensuremath{\hbar}/4\ensuremath{\pi}{k}_{B}

bound by a factor of about

1.4

DOI: 10.1103/physreva.89.023621

Related papers

$η/s$ of the Normal Phase of Unitary Fermi Gas from $\varepsilon$ Expansion$η/s$ of the Normal Phase of Unitary Fermi Gas from $\varepsilon$ Expansion

Andrei B. Kryjevski

Using $\varepsilon$ -expansion technique we compute $η/s$ , where $η$ is the shear viscosity, $s$ is the entropy density, of the normal phase of unitary Fermi gas in $d=4-\varepsilon$ dimensions to LO in $\varepsilon$ . We use kinetic theory approach and solve transport equations for medium perturbed by a shear hydrodynamic flow. The collision integrals are calculated to $\varepsilon^2$ which is LO. The LO result is temperature independent with $η/\rm s\simeq (0.11/\varepsilon^2)(\hbar/k_B).$ The $d=3$ prediction for $η/\rm s$ exceeds the $\hbar/4 πk_B$ bound by a factor of about $1.4.$ Using $\varepsilon$-expansion technique we compute $η/s$, where $η$ is the shear viscosity, $s$ is the entropy density, of the normal phase of unitary Fermi gas in $d=4-\varepsilon$ dimensions to LO in $\varepsilon$. We use kinetic theory approach and solve transport equations for medium perturbed by a shear hydrodynamic flow. The collision integrals are calculated to $\varepsilon^2$ which is LO. The LO result is temperature independent with $η/\rm s\simeq (0.11/\varepsilon^2)(\hbar/k_B).$ The $d=3$ prediction for $η/\rm s$ exceeds the $\hbar/4 πk_B$ bound by a factor of about $1.4.$

arXiv:1206.0059Jun 1, 20120 citations

Determination of the Density and Temperature Dependence of the Shear Viscosity of a Unitary Fermi Gas Based on Hydrodynamic Flow

Marcus Bluhm, Jiaxun Hou, Thomas Schäfer

We determine the shear viscosity of the ultracold Fermi gas at unitarity in the normal phase using hydrodynamic expansion data. The analysis is based on a generalized fluid dynamic framework which ensures a smooth transition between the fluid dynamic core of the cloud and the ballistic corona. We use expansion data taken by Joseph, Elliott, and Thomas [Shear Viscosity of a Universal Fermi Gas Near the Superfluid Phase Transition, Phys. Rev. Lett. 115, 020401 (2015).PRLTAO0031-900710.1103/PhysRevLett.115.020401] and measurements of the equation of state by Ku et al. [Revealing the superfluid lambda transition in the universal thermodynamics of a unitary Fermi gas, Science 335, 563 (2012).SCIEAS0036-807510.1126/science.1214987]. We find that the shear viscosity to particle density ratio just above the critical temperature is η/n|_{T_{c}}=0.41±0.11. We also obtain evidence that the shear viscosity to entropy density ratio has a minimum slightly above T_{c} with η/s|_{min}=0.50±0.10.We determine the shear viscosity of the ultracold Fermi gas at unitarity in the normal phase using hydrodynamic expansion data. The analysis is based on a generalized fluid dynamic framework which ensures a smooth transition between the fluid dynamic core of the cloud and the ballistic corona. We use expansion data taken by Joseph, Elliott, and Thomas [Shear Viscosity of a Universal Fermi Gas Near the Superfluid Phase Transition, Phys. Rev. Lett. 115, 020401 (2015).PRLTAO0031-900710.1103/PhysRevLett.115.020401] and measurements of the equation of state by Ku et al. [Revealing the superfluid lambda transition in the universal thermodynamics of a unitary Fermi gas, Science 335, 563 (2012).SCIEAS0036-807510.1126/science.1214987]. We find that the shear viscosity to particle density ratio just above the critical temperature is η/n|_{T_{c}}=0.41±0.11. We also obtain evidence that the shear viscosity to entropy density ratio has a minimum slightly above T_{c} with η/s|_{min}=0.50±0.10.

arXiv:1704.03720Physical Review LettersAug 10, 201736 citations

Nearly Perfect Fluidity in the Dilute Fermi Gas: An Update

Thomas Schäfer

Aug 16, 20160 citations

Viscosity and scale invariance in the unitary Fermi gas

Tilman Enss, R. Haussmann, W. Zwerger

We compute the shear viscosity of the unitary Fermi gas above the superfluid transition temperature, using a diagrammatic technique that starts from the exact Kubo formula. The formalism obeys a Ward identity associated with scale invariance which guarantees that the bulk viscosity vanishes identically. For the shear viscosity, vertex corrections and the associated Aslamazov-Larkin contributions are shown to be crucial to reproduce the full Boltzmann equation result in the high-temperature, low fugacity limit. The frequency dependent shear viscosity $\eta(\omega)$ exhibits a Drude-like transport peak and a power-law tail at large frequencies which is proportional to the Tan contact. The weight in the transport peak is given by the equilibrium pressure, in agreement with a sum rule due to Taylor and Randeria. Near the superfluid transition the peak width is of the order of $0.5 T_F$ , thus invalidating a quasiparticle description. The ratio $\eta/s$ between the static shear viscosity and the entropy density exhibits a minimum near the superfluid transition temperature whose value is larger than the string theory bound $\hbar/(4\pi k_B)$ by a factor of about seven.We compute the shear viscosity of the unitary Fermi gas above the superfluid transition temperature, using a diagrammatic technique that starts from the exact Kubo formula. The formalism obeys a Ward identity associated with scale invariance which guarantees that the bulk viscosity vanishes identically. For the shear viscosity, vertex corrections and the associated Aslamazov-Larkin contributions are shown to be crucial to reproduce the full Boltzmann equation result in the high-temperature, low fugacity limit. The frequency dependent shear viscosity $\eta(\omega)$ exhibits a Drude-like transport peak and a power-law tail at large frequencies which is proportional to the Tan contact. The weight in the transport peak is given by the equilibrium pressure, in agreement with a sum rule due to Taylor and Randeria. Near the superfluid transition the peak width is of the order of $0.5 T_F$, thus invalidating a quasiparticle description. The ratio $\eta/s$ between the static shear viscosity and the entropy density exhibits a minimum near the superfluid transition temperature whose value is larger than the string theory bound $\hbar/(4\pi k_B)$ by a factor of about seven.

Jan 20, 2016164 citations

Hydrodynamic fluctuations and the minimum shear viscosity of the dilute Fermi gas at unitarity

Clifford Chafin, Thomas Schäfer

We study hydrodynamic fluctuations in a nonrelativistic fluid. We show that in three dimensions, fluctuations lead to a minimum in the shear viscosity to entropy density ratio $\ensuremath{\eta}/s$ as a function of the temperature. The minimum provides a bound on $\ensuremath{\eta}/s$ which is independent of the conjectured bound in string theory, $\ensuremath{\eta}/s\ensuremath{\ge}\ensuremath{\hbar}/(4\ensuremath{\pi}{k}_{B})$, where $s$ is the entropy density. For the dilute Fermi gas at unitarity, we find $\ensuremath{\eta}/s\ensuremath{\gtrsim}0.2\ensuremath{\hbar}$. This bound is not universal---it depends on the thermodynamic properties of the unitary Fermi gas and on empirical information about the range of validity of hydrodynamics. We also find that the viscous relaxation time of a hydrodynamic mode with frequency $\ensuremath{\omega}$ diverges as $1/\sqrt{\ensuremath{\omega}}$, and that the shear viscosity in two dimensions diverges as $\mathrm{ln}(1/\ensuremath{\omega})$.We study hydrodynamic fluctuations in a nonrelativistic fluid. We show that in three dimensions, fluctuations lead to a minimum in the shear viscosity to entropy density ratio $\ensuremath{\eta}/s$ as a function of the temperature. The minimum provides a bound on $\ensuremath{\eta}/s$ which is independent of the conjectured bound in string theory, $\ensuremath{\eta}/s\ensuremath{\ge}\ensuremath{\hbar}/(4\ensuremath{\pi}{k}_{B})$, where $s$ is the entropy density. For the dilute Fermi gas at unitarity, we find $\ensuremath{\eta}/s\ensuremath{\gtrsim}0.2\ensuremath{\hbar}$. This bound is not universal---it depends on the thermodynamic properties of the unitary Fermi gas and on empirical information about the range of validity of hydrodynamics. We also find that the viscous relaxation time of a hydrodynamic mode with frequency $\ensuremath{\omega}$ diverges as $1/\sqrt{\ensuremath{\omega}}$, and that the shear viscosity in two dimensions diverges as $\mathrm{ln}(1/\ensuremath{\omega})$.

arXiv:1209.1006Physical Review AFeb 27, 201351 citations

Medium effects and the shear viscosity of the dilute Fermi gas away from the conformal limit

Marcus Bluhm, Thomas Schäfer

We study the shear viscosity of a dilute Fermi gas as a function of the scattering length in the vicinity of the unitarity limit. The calculation is based on kinetic theory, which provides a systematic approach to transport properties in the limit in which the fugacity $z=n{\ensuremath{\lambda}}^{3}/2$ is small. Here, $n$ is the density of the gas and $\ensuremath{\lambda}$ is the thermal wavelength of the fermions. At leading order in the fugacity expansion, the shear viscosity is independent of density, and the minimum shear viscosity is achieved at unitarity. At the next order, medium effects modify the scattering amplitude as well as the quasiparticle energy and velocity. We show that these effects shift the minimum of the shear viscosity to the Bose-Einstein condensation side of the resonance, in agreement with the result of recent experiments.We study the shear viscosity of a dilute Fermi gas as a function of the scattering length in the vicinity of the unitarity limit. The calculation is based on kinetic theory, which provides a systematic approach to transport properties in the limit in which the fugacity $z=n{\ensuremath{\lambda}}^{3}/2$ is small. Here, $n$ is the density of the gas and $\ensuremath{\lambda}$ is the thermal wavelength of the fermions. At leading order in the fugacity expansion, the shear viscosity is independent of density, and the minimum shear viscosity is achieved at unitarity. At the next order, medium effects modify the scattering amplitude as well as the quasiparticle energy and velocity. We show that these effects shift the minimum of the shear viscosity to the Bose-Einstein condensation side of the resonance, in agreement with the result of recent experiments.

arXiv:1410.2827Physical Review ADec 8, 201411 citations

Using the

\ensuremath{\varepsilon}

-expansion technique, we compute

\ensuremath{\eta}/s

, where

\ensuremath{\eta}

is the shear viscosity and

s

is the entropy density of the normal phase of unitary Fermi gas in

d=4\ensuremath{-}\ensuremath{\varepsilon}

dimensions to leading order (LO) in

\ensuremath{\varepsilon}

. We use the kinetic theory approach and solve transport equations for a medium perturbed by a shear hydrodynamic flow. The collision integrals are calculated to

{\ensuremath{\varepsilon}}^{2}

, which is LO. The LO result is temperature independent with

\ensuremath{\eta}/s\ensuremath{\simeq}(0.11/{\ensuremath{\varepsilon}}^{2})(\ensuremath{\hbar}/{k}_{B})

. The

d=3

prediction for

\ensuremath{\eta}/s

exceeds the

\ensuremath{\hbar}/4\ensuremath{\pi}{k}_{B}

bound by a factor of about

1.4

η/s

of the Normal Phase of Unitary Fermi Gas from

\varepsilon

Expansion

Using

\varepsilon

-expansion technique we compute

η/s

, where

η

is the shear viscosity,

s

is the entropy density, of the normal phase of unitary Fermi gas in

d=4-\varepsilon

dimensions to LO in

\varepsilon

. We use kinetic theory approach and solve transport equations for medium perturbed by a shear hydrodynamic flow. The collision integrals are calculated to

\varepsilon^2

which is LO. The LO result is temperature independent with

η/\rm s\simeq (0.11/\varepsilon^2)(\hbar/k_B).

The

d=3

prediction for

η/\rm s

exceeds the

\hbar/4 πk_B

bound by a factor of about

1.4.

\eta(\omega)

exhibits a Drude-like transport peak and a power-law tail at large frequencies which is proportional to the Tan contact. The weight in the transport peak is given by the equilibrium pressure, in agreement with a sum rule due to Taylor and Randeria. Near the superfluid transition the peak width is of the order of

0.5 T_F

, thus invalidating a quasiparticle description. The ratio

\eta/s

between the static shear viscosity and the entropy density exhibits a minimum near the superfluid transition temperature whose value is larger than the string theory bound

\hbar/(4\pi k_B)

by a factor of about seven.

s

is the entropy density. For the dilute Fermi gas at unitarity, we find $\ensuremath{\eta}/s\ensuremath{\gtrsim}0.2\ensuremath{\hbar}$. This bound is not universal---it depends on the thermodynamic properties of the unitary Fermi gas and on empirical information about the range of validity of hydrodynamics. We also find that the viscous relaxation time of a hydrodynamic mode with frequency $\ensuremath{\omega}$ diverges as $1/\sqrt{\ensuremath{\omega}}$, and that the shear viscosity in two dimensions diverges as $\mathrm{ln}(1/\ensuremath{\omega})$.

n

is the density of the gas and $\ensuremath{\lambda}$ is the thermal wavelength of the fermions. At leading order in the fugacity expansion, the shear viscosity is independent of density, and the minimum shear viscosity is achieved at unitarity. At the next order, medium effects modify the scattering amplitude as well as the quasiparticle energy and velocity. We show that these effects shift the minimum of the shear viscosity to the Bose-Einstein condensation side of the resonance, in agreement with the result of recent experiments.